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This is not the right way to think about it, these are just the equations of motion in a different language. The right way is the path integral, and these equations give you differential identities in the path integral, but you renormalize the path integral itself, not the equations. You don't "solve equations" in quantum field theory, you sum over all possible states.

As explained on Wikipedia, Schwinger's action principle is just a formulation of the path integral which included fermions before Grassman integration was formulate by Candlin. It became obsolete in 1956, after Grassman integration defined the path integral for Fermi fields. There is no gain in thinking about the differential equations, since these equations are not classical things for classical functions, they are operator equations which give you constraints on operator correlations.